Shaped beam pattern synthesis with desired nulling level and minimum sidelobe level

The wide-beam pattern via the array antenna has been widely used in receiving satellite multimedia signals with mobile ground platforms, such as cars, trains, and ships [1−4]. For such applications, the array antenna platforms would maneuver fast, changing their positions and directions. Hence, a wide-beam mainlobe is required to ensure that the beam would always point to the satellite all the time. Meanwhile, the signal of interest would be rather weak after long-distance transmission. Therefore, it is necessary to synthesize a deep nulling region for the anti-interference purpose [5−7] and suppress the sidelobe level (SLL) as much as possible for the denoise purpose [8−10]. To realize such an objective, this paper discusses the suppression of SLL in a wide-beam pattern with a low nulling level (NL) in the nulling region.

To minimize SLL of a shaped-beam pattern, such as a flat-top wide-beam pattern, the most used approach is solving the shaped-beam pattern synthesis (SBPS) problem [11−13]. The SBPS-based approaches optimize the beam pattern to minimize SLL, which focuses on synthesizing a predetermined beam shape. By this means, the power gain in the mainlobe is not the variable to be optimized. However, the power gain is an important factor in such types of applications. For wide-beam applications, the SBPS-based approaches cannot guarantee achieving the highest power gain in the mainlobe. In fact, several works on solving the power gain pattern synthesis (PGPS) problem have proven that the power gain optimization for wide beams can generate a higher gain in the mainlobe [14−16]. Conversely, the obtained SLL would not be the lowest if a certain power gain is required. Under such a situation, it is reasonable to infer that the PGPS-based approaches have the potential to suppress SLL while generating the same gain as the SBPS-based approaches.

In our previous work, the further suppression of SLL of a wide-beam pattern with the desired low NL nulling region has been discussed by solving the PGPS problem [17]. The numerical simulations indicate that the PGPS-based approach can suppress SLL by more than 3 dB. However, Ref. [17] solves the non-convex PGPS problem by transforming the complex-valued mainlobe pattern into its real part, which would unavoidably deteriorate the performance. The PGPS problem has a general Rayleigh Quotient form with the array excitation being the optimized variable. In this paper, the non-convex PGPS problem is solved in a smoother way by transforming the primal problem into convex sub-problems. By assuming the left array excitation is the known priori, the primal problem is convex with respect to the right excitation. In such a way, it can be solved iteratively by updating the left-side/right-side array excitation alternatively, resulting in a better array excitation which would further suppress SLL while obtaining desired NL and the same mainlobe power gain as the SBPS-based approach. Two different patterns, including the isotropic element pattern (IEP) array and the active element pattern (AEP) array, are simulated to assess the proposed algorithm by comparing with the existing algorithms.

The rest of this paper are organized as: Section 2 formulates the PGPS problem to be solved, Section 3 deduces the proposed algorithm, Section 4 simulates and remarks the numerical examples, and Section 5 draws the conclusions.

Consider an arbitrarily distributed N-element linear array, and denote the element positions as $ {r_n} $ $(n = 1{\rm{,}}\;2{\rm{,}}\; \cdots{\rm{,}}\;N)$, the extension to the planar array is straightforward. Denote $ {\omega _n} $ as the array excitation of the nth element, the array far-field electric field at $ \theta $ can be expressed as

where

where $ \theta $ is the beam direction; $ E(\theta ) $ is the far-field electric field of the array; w and a are the array excitation and the array factor with $ {a_n}(\theta ) = \exp (2{\rm{j}}\pi \kappa {r_n}\sin \theta ) $, respectively; $ \kappa = {\lambda \mathord{\left/ {\vphantom {\lambda {2\pi }}} \right. } {2\pi }} $ is the electromagnetic wavenumber. If consider the element total efficiency, the effective radiation beam pattern can be expressed as $ {\left( {{\bf{w}} \odot \sqrt \eta } \right)^{\rm{H}}}{\bf{a}}\left( \theta \right) $ [15], where $ \odot $ is element-wise multiplication of two vectors, and the element total efficiency including the reflection, conduction, and dielectric efficiencies [18]. In practice, the total efficiency is affected by both the array weight and beam direction. However, the relationship among $ \eta $, the array excitation, and the beam direction can be computed experimentally, where $ \eta $ is assumed as the known priori in this paper. To simplify the expression, mark $ {\bf{w}} \odot \sqrt \eta $ as w. With a desired low NL nulling region, a typical SBPS problem to minimize SLL is

s.t.

where $ \varepsilon $ is the mainlobe ripple level (MRL), $ \rho $ is SLL, $ \gamma $ is NL, $ {\Theta _{{\rm{ML}}}} $ is the wide-beam mainlobe region, $ {\Theta _{{\rm{SL}}}} $ is the sidelobe region, $ {\Theta _{{\rm{Null}}}} $ is the nulling region, $ {\theta _{\rm{M}}} $ is the angle in $ {\Theta _{{\rm{ML}}}} $, $ {\theta _{\rm{S}}} $ is the angle in $ {\Theta _{{\rm{SL}}}} $, $ {\theta _{{\rm{Nu}}}} $ is the angle in $ {\Theta _{{\rm{Null}}}} $, $ {f_d}\left( \theta \right) $ is the desired pattern, and $ \left\| \cdot \right\|_2^{} $ is the l₂ norm. Given desired MRL and NL, this problem tries to minimize SLL. For a flat-top wide beam, there is

In such a form, (3) is convex and can be converted into a semi-definite programming form [19]. Hence, it can be solved with the existing reliable convex problem solvers, including SeDumi [20] and CVX [21] .

Given the radiation pattern, the array power gain is defined as [16,22]

where ${{\bf{{A}}}_\theta } = {{\bf{a}}^{\rm{H}}}\left( \theta \right){\bf{a}}\left( \theta \right)$, ${\bf{{A}}} = \oiint_{\left( {\theta {\rm{,}}\phi } \right)} {{{\bf{a}}^{\rm{H}}}\left( \theta \right){\bf{a}}\left( \theta \right)\sin \left( \theta \right){\rm{d}}\theta } {\rm{d}}\phi$, and $\left( {\theta {\rm{,}}\;\phi } \right)$ represents the angle in the spatial region. These two parameters can be treated as known ones given the array layout. To further suppress SLL, the PGPS problem which optimizes the power gain pattern directly can be described as follows:

s.t.

where $ {G_0} $ is the minimum array power gain in the mainlobe $ {\Theta _{{\rm{ML}}}} $, which is computed via the array excitation obtained by solving (3). Notice that $ {{\bf{w}}^{\rm{H}}}{\bf{Aw}} $ > 0 for any $ {\bf{w}} \ne $ 0, so the matrix $ {\bf{A}} $ is positive definite. From its definition, A is the Hermitian symmetric, resulting in the matrix $ {\bf{A}} $ being the Hermitian positive definite. Therefore, there must be a positive definite matrix C which satisfies A = C^HC. Denote $ {{\bf{P}}_\theta }{\rm{ = }}{{\bf{C}}^{ - {\rm{H}}}}{{\bf{A}}_\theta }{{\bf{C}}^{ - 1}} $ and $ {\bf{x}} = {\bf{Cw}} $, the primal optimization problem (6) is equivalent to

s.t.

The zooming of x would enlarge/lessen the denominator/numerator of the Rayleigh Quotient in the above constraints simultaneously. As a result, the variation of the x’s modulus would not affect the final solution to the above optimization problem. Set $ {\left\| {\bf{x}} \right\|_2} = 1 $, the above optimization problem is converted as

s.t.

Since the first constraint is not convex, the optimal solution to the above optimization problem can be obtained by searching the whole solution space, which is intractable even with a small size of the array. To overcome this shortcoming, Ref. [17] converted the first constraint into a convex one with 2real $ \left( {{{\bf{x}}^{\rm{H}}}{{\bf{P}}_{{\theta _{\rm{M}}}}}{\bf{x}}} \right) \ge {G_0} $. Besides, the constraint $ {\left\| {\bf{x}} \right\|_2}{\rm{ = }}1 $ can be recast as $ {\left\| {\bf{x}} \right\|_2} \le 1 $ in this minimization optimization problem. Hence, (8) can be approximated by

s.t.

where real(·) represents the real part of a complex-valued number, $ {g_0} = \sqrt {{G_0}/2} $, and the primal optimization problem is replaced by an easily solvable convex one.

In this section, an iterative algorithm is proposed by iteratively solving the left-side and right-side array excitations x. Mark the left-side/right-side excitation as $ {{\bf{x}}_l} $ and $ {{\bf{x}}_r} $, respectively. The problem can be rewritten as

s.t.

Once $ {{\bf{x}}_l} $ or $ {{\bf{x}}_r} $ is fixed, the above problem is recast into a convex linear programming problem without considering the last constraint, i.e., $ {{\bf{x}}_l} = {{\bf{x}}_r} $. Notice that the power gain is always real-valued, the first constraint is written as 2real $ \left( {{\bf{x}}_l^{\rm{H}}{{\bf{P}}_{{\theta _{\rm{M}}}}}{{\bf{x}}_r}} \right) \ge {G_0} $ to ensure its convexity, while the latter two constraints are written as $ \left| {{\bf{x}}_l^{\rm{H}}{{\bf{P}}_{{\theta _{\rm{S}}}}}{{\bf{x}}_r}} \right| \le \rho {G_0} $ and $ \left| {{\bf{x}}_l^{\rm{H}}{{\bf{P}}_{{\theta _{{\rm{Nu}}}}}}{{\bf{x}}_r}} \right| \le \gamma {G_0} $. By omitting the equality constraint, i.e., $ {{\bf{x}}_l} = {{\bf{x}}_r} $, the above problem can be rewritten as a series of following sub-problems:

s.t.

The above problem is convex, if all its constraints are convex. Hence, its optimal solution can be obtained with the existing convex problem solvers. It is necessary to note that the optimal solution, i.e. $ {{\bf{x}}_r} $, to (11) is not the solution to (10), since it cannot guarantee that the obtained $ {{\bf{x}}_r} $ equals the initialized $ {{\bf{x}}_l} $. To realize the constraint $ {{\bf{x}}_l} = {{\bf{x}}_r} $, the above problem should be solved repeatedly. After obtaining $ {{\bf{x}}_r} $, $ {{\bf{x}}_l} $ can be updated accordingly and then repeated to solve (11). In such a manner, the proposed algorithm would approximate $ {{\bf{x}}_l} $ to the optimal solution gradually. The proposed algorithm is summarized in Algorithm 1. The proposed algorithm is terminated, when $ {{\bf{x}}_l} $ is lose to $ {{\bf{x}}_r} $ or when $ {{\bf{x}}_r} $ cannot be renewed.

Algorithm 1: The proposed algorithm for minimizing SLL of a wide-beam pattern with desired low NL.

1. Initialize $ {{\bf{x}}_l} $, ${\rm{er}}{{\rm{r}}_0} \in \left[ {{{10}^{ - 6}} {\rm{,}} \;{{10}^{ - 3}}} \right]$ , $\alpha \in \left[ {0.2 {\rm{,}} \;0.5} \right]$ , and $\delta \in \left[ {0.005 {\rm{,}} \;0.05} \right]$.

2. Obtain $ {{\bf{A}}_\theta } $ and $ {\bf{A}} $ via (5).

3. Calculate $ {G_0} $ by solving (3).

4. While $ {\rm{err > er}}{{\rm{r}}_0} $ and $ \alpha > 0 $, do

5. Solve (11) to obtain $ {{\bf{x}}_r} $.

6. $ {{\bf{x}}_l} = \left( {0.5 + \alpha } \right){{\bf{x}}_r} + \left( {0.5 - \alpha } \right){{\bf{x}}_l} $.

7. $ \alpha = \alpha - \delta $.

8. $ {\rm{err = }}\left| {{{\bf{x}}_l} - {{\bf{x}}_r}} \right| $.

9. End while.

10. Return $ {\bf{w}} = {{\bf{C}}^{ - 1}}{{\bf{x}}_l} $.

With both IEP and AEP arrays, this section simulates several numerical examples to validate the effectiveness of the proposed algorithm. IEP has an omnidirectional radiation pattern. AEP is simulated via High Frequency Structure Simulator (HFSS) by exciting one array element while connecting the others with the matching node [23,24]. In this way, AEP would consider the mutual coupling of array element. For the comparison purpose, two existing algorithms are compared:

1) The SBPS-based algorithm which solves (3).

2) The existing PGPS-based algorithm which solves (9). $ {G_0} $ in a wide-beam mainlobe is obtained with the optimal w via the compared SBPS-based algorithm [17].

The initialization $ {{\bf{w}}_l} $ $ \left( {{{\bf{x}}_l} = {\bf{C}}{{\bf{w}}_l}} \right) $ for the proposed algorithm is set as

where $ \Delta \omega $ is decided by the beamwidth and $ {\theta _c} $ is the beam center. Notice that such an initialization might cause (11) unsolvable in the first several iterations, we initialize $ {{\bf{w}}_l} = {\omega _n}\left( {1 - {c_0}} \right) + {c_0}{{{\tilde {\bf w}}}_n} $, where $ {{{\tilde {\bf w}}}_n} $ is the array excitation via [17] and ${c_0} \in \left[ {0.5{\rm{,}}{\rm{ }}0.9} \right]$ is an adjustable predetermined parameter.

A uniformly distributed 32-element linear array is simulated. Different wide-beam mainlobes, including the mainlobe beamwidth ${\Theta _{{\rm{bw}}}} = $ 30°, 40°, and 50° are tested. Set $ \varepsilon = 0.1 $ for MRL in the SBPS-based algorithm, $ \alpha = 0.2 $, $ \delta = 0.01 $, and $ {\rm{er}}{{\rm{r}}_0} = $ $ {10^{ - 3}} $ for the proposed algorithm. Set the nulling region, $ {\Theta _{{\rm{Null}}}} \in $ $\left[ {{{40}\circ }{\rm{, }}{{60}\circ }} \right]$ with desired NL being $ \gamma = - 80 $ dB. The mainlobe regions are set as ${\Theta _{{\rm{ML}}}} \in \left[ { - {{15}\circ }{\rm{,}}{\rm{ }}{{15}\circ }} \right]$, $\left[ { - {{20}\circ }{\rm{,}}{\rm{ }}{{20}\circ }} \right]$, and $\left[ { - {{25}\circ }{\rm{,}}{\rm{ }}{{25}\circ }} \right]$. The minimum angular distances between the mainlobe region and the sidelobe region, and between the sidelobe region and the nulling region are $ {5\circ } $. Hence, the sidelobe regions are ${\Theta _{{\rm{SL}}}} \in \left[ { - {{90}\circ }{\rm{,}}{\rm{ }}{{70}\circ }} \right] \cup \left[ {{{20}\circ }{\rm{,}}{\rm{ }}{{35}\circ }} \right] \cup \left[ {{{65}\circ }{\rm{,}}{\rm{ }}{{90}\circ }} \right]$, ${\Theta _{{\rm{SL}}}} \in \left[ { - {{90}\circ }{\rm{,}}{\rm{ }}{{65}\circ }} \right] \cup \left[ {{{25}\circ }{\rm{,}}{\rm{ }}{{35}\circ }} \right] \cup \left[ {{{65}\circ }{\rm{,}}{\rm{ }}{{90}\circ }} \right]$, and ${\Theta _{{\rm{SL}}}} \in \left[ { - {{90}\circ }{\rm{,}}{\rm{ }}{{66}\circ }} \right] \cup \left[ {{{30}\circ }{\rm{,}}{\rm{ }}{{35}\circ }} \right] \cup \left[ {{{65}\circ }{\rm{,}}{\rm{ }}{{90}\circ }} \right]$, respectively. SLL is defined as the gap of the maximum power gain in $ {\Theta _{{\rm{SL}}}} $ to the minimum power gain in $ {\Theta _{{\rm{ML}}}} $.

Fig. 1 plots out the synthesized results via different compared algorithms for the beam center ${\theta _c} = {0\circ }$. It shows the proposed algorithm can obtain the lowest SLL while generating the same power gain in $ {\Theta _{{\rm{ML}}}} $ and the same NL in $ {\Theta _{{\rm{Null}}}} $. It also indicates that the PGPS-based algorithm reported in Ref. [17] obtains lower SLL than the SBPS-based algorithm. All algorithms obtain the same $ {G_0} $ in $ {\Theta _{{\rm{ML}}}} $ which are 4.98 dBi, 3.89 dBi, and 3.02 dBi, respectively, for $ \gamma = - 80 $ dB and ${\Theta _{{\rm{bw}}}} = {50\circ }$. NL for the SBPS-based algorithm, the PGPS-based algorithm reported in Ref. [17], and the proposed algorithm are averagely −79.6 dB, −80.0 dB, and −80.0 dB, respectively. The SBPS-based algorithm cannot obtain strictly −80-dB NL, this is caused by the existence of the mainlobe ripple. According to the optimization model for (3), its SLL is decided as the mainlobe ripple center minus the peak sidelobe level, in this sense, it is reasonable to conclude that the SBPS-based algorithm can realize desired NL.

Figure 1.  Synthesized power gain patterns for different beamwidth via linear IEP array antenna with $ {\theta _c} = {0\circ } $ and $ \gamma = - 80 $ dB: (a) $ {\Theta _{{\text{bw}}}}{\text{ = }}{30\circ } $, (b) $ {\Theta _{{\text{bw}}}}{\text{ = }}{40\circ } $, and (c) $ {\Theta _{{\text{bw}}}}{\text{ = }}{50\circ } $.

To show the comparisons clearer, Table 1 provides the specific obtained SLLs for different beamwidth. The average SLLs for the two compared algorithms are −34.4 dB and −40.5 dB, respectively. Hence, the proposed algorithm suppresses SLL by about 17.7 dB and 11.6 dB, respectively, compared with the SBPS-based algorithm and the PGPS-based algorithm in Ref. [17]. Moreover, the PGPS-based algorithm suppresses SLL by about 6.1 dB over the compared SBPS-based algorithm. This example proves that the proposed algorithm is rather competitive to suppress SLL, when synthesizing an array beam pattern with desired $ {G_0} $ in $ {\Theta _{{\rm{ML}}}} $ and desired NL in $ {\Theta _{{\rm{Null}}}} $.

Fig. 2 shows the synthesized results, when the beam center is $ {\theta _c} = {10\circ } $ for $ {\Theta _{{\rm{bw}}}}{\rm{ = }}{30\circ } $ and $ {\Theta _{{\rm{bw}}}}{\rm{ = }} $ $ {40\circ } $. All algorithms obtain the same minimum power gain, i.e., 4.95 dBi for $ {\Theta _{{\rm{bw}}}}{\rm{ = }}{30\circ } $ and 3.71 dBi for $ {\Theta _{{\rm{bw}}}}{\rm{ = }}{40\circ } $, while generating desired –80-dB NL. Their SLLs are $\left\{ { - 29.6\;{\rm{dB}}{\rm{,}}{\rm{ }} - 35.2\;{\rm{dB}}{\rm{,}}{\rm{ }} - 49.0\;{\rm{dB}}} \right\}$ for $ {\Theta _{{\rm{bw}}}}{\rm{ = }}{30\circ } $ and $\left\{ { - 11.1\;{\rm{dB}}{\rm{,}}\;\left. { - 44.8\;{\rm{dB}}{\rm{, }}- 63.1\;{\rm{dB}}} \right\}} \right.$ for $ {\Theta _{{\rm{bw}}}} = {40\circ } $. The results validate that the proposed algorithm outperforms the compared SBPS-based/PGPS-based algorithms. Averagely, the proposed algorithm is able to further suppress SLL by more than 30 dB and 15 dB, respectively. Compared with the results in Fig. 1, the advantage of the proposed algorithm is more notable when the mainlobe is not broadside-axis symmetric.

Figure 2.  Synthesized power gain patterns for different beamwidth via linear IEP array antenna with $ {\theta _c} = {10\circ } $ and $ \gamma = - 80 $ dB: (a) $ {\Theta _{{\text{bw}}}} = {30\circ } $ and (b) $ {\Theta _{{\text{bw}}}}{\text{ = }}{40\circ } $.

Fig. 3 draws the obtained SLLs for different desired NLs, i.e., $\gamma \in \left[ { - 100\;{\rm{dB}}{\rm{,}}{\rm{ }} - 60\;{\rm{dB}}} \right]$. The beamwidth is ${\Theta _{{\rm{bw}}}} = {20\circ }$ with the beam center being ${\theta _c} = {10\circ }$. It demonstrates that the proposed algorithm always obtains the lowest SLL than the compared SBPS-based/PGPS-based algorithms. Specifically, the average obtained SLL is about −37.3 dB, −39.6 dB, and −52.0 dB, respectively. Hence, the proposed algorithm suppresses SLL by about −15.0 dB and −12.4 dB, when compared with the SBPS-based algorithm and the PGPS-based algorithm [17], respectively. Fig. 3 also illustrates that lower NL would cause higher SLL, for instance, the obtained SLLs are $\{ - 35.6{\rm{ dB}}{\rm{,}}{\rm{ }} - 38.3{\rm{ dB}}{\rm{,}} - 50.4{\rm{ dB}} \}$ at $\gamma = - 80 \; {\rm{dB}}$ and $\left\{ { - 38.4{\rm{ dB}}{\rm{,}}} \right.$ $\left. { - 40.9{\rm{ dB}}{\rm{,}}{\rm{ }} - 53.7{\rm{ dB}}} \right\}$ at $ \gamma = - 60 $ dB.

Figure 3.  Synthesized power gain patterns for different NLs, i.e. $\gamma \in $ [−100 dB, −60 dB], at the nulling region of [40°, 60°] via linear IEP array antenna.

A half-wavelength uniformly distributed origin-symmetric 41-element linear array is simulated via HFSS to obtain AEPs. With such radiation patterns, the mutual coupling of array elements is also considered. Fig. 4 shows AEPs of the simulated array antenna. It illustrates that the radiation pattern is not isotropic and the 3-dB beamwidth is larger than 90°.

Figure 4.  Normalized AEPs of a half-wavelength uniformly distributed origin-symmetric 41-element linear array simulated via HFSS.

Different wide-beam mainlobes, including the mainlobe beamwidth $ {\Theta _{{\rm{bw}}}} = {20\circ } $, $ {30\circ } $, and $ {40\circ } $ are tested. Set the nulling region ${\Theta _{{\rm{Null}}}} \in \left[ {{{50}\circ }{\rm{,}}{\rm{ }}{{60}\circ }} \right]$ with desired NL being $ \gamma = - 80 $ dB. The mainlobe regions are set as $ {\Theta _{{\rm{ML}}}} \in $ $\left[ { - {{10}\circ }{\rm{, }}{{10}\circ }} \right]$, $\left[ { - {{15}\circ }{\rm{,}}{\rm{ }}{{15}\circ }} \right]$, and $\left[ { - {{20}\circ }{\rm{,}}{\rm{ }}{{20}\circ }} \right]$. The minimum angular distances between $ {\Theta _{{\rm{ML}}}} $ and $ {\Theta _{{\rm{SL}}}} $, and between the sidelobe region and the nulling region are ${3\circ }$. Hence, the sidelobe regions are ${\Theta _{{\rm{SL}}}} \in \left[ { - {{90}\circ }{\rm{,}}{\rm{ }}{{77}\circ }} \right] \cup \left[ {{{13}\circ }{\rm{,}}{\rm{ }}{{47}\circ }} \right] \cup \left[ {{{63}\circ }{\rm{,}}{\rm{ }}{{90}\circ }} \right]$, ${\Theta _{{\rm{SL}}}} \in \left[ - {{90}\circ }{\rm{,}} \right. \left. {\rm{ }}{{72}\circ } \right] \cup \left[ {{{18}\circ }{\rm{,}}{\rm{ }}{{47}\circ }} \right] \cup \left[ {{{63}\circ }{\rm{,}}{\rm{ }}{{90}\circ }} \right]$, and $ {\Theta _{{\rm{SL}}}} \in $ $\left[ { - {{90}\circ }{\rm{,}}{\rm{ }}{{67}\circ }} \right] \cup \left[ {{{23}\circ }{\rm{,}}{\rm{ }}{{47}\circ }} \right] \cup \left[ {{{63}\circ }{\rm{,}}{\rm{ }}{{90}\circ }} \right]$.

Fig. 5 shows the synthesized power gain patterns of the compared algorithms. It validates that the proposed algorithm always obtains much lower SLL than the compared SBPS-based/PGPS-based algorithms, when achieves the same $ {G_0} $ in $ {\Theta _{{\rm{ML}}}} $ and –80-dB NL in $ {\Theta _{{\rm{Null}}}} $. The minimum power gain $ {G_0} $ in the wide-beam mainlobe is {2.59 dBi, 1.75 dBi, 1.05 dBi} for ${\Theta _{{\rm{bw}}}} = \left\{ {{{20}\circ }{\rm{,}}{\rm{ }}{{30}\circ }{\rm{,}}{\rm{ }}{{40}\circ }} \right\}$. To show the comparisons clearer, Table 2 provides the specific obtained SLLs for different beamwidth. Average SLLs for these compared algorithms are −4.1 dB, −15.9 dB, and −61.8 dB, respectively. Hence, the proposed algorithm suppresses SLL by about 56.7 dB and 45.9 dB compared with the SBPS-based algorithm and the PGPS-based one reported in Ref. [17], respectively. Moreover, the PGPS-based algorithm in Ref. [17] can further suppress SLL by about 11.7 dB over the compared SBPS-based algorithm. The obtained power gains, i.e., $ {G_0} $, in the mainlobe are lower than those in IEP scenarios. Because the radiation pattern of AEPs is not omnidirectional and more power is radiated into the “stopband” region between the mainlobe region and the sidelobe region. However, the same remark as in subsection 3.1 can be drawn that the proposed algorithm outperforms the compared SBPS-based/PGPS-based algorithm on minimizing SLL to achieve the same power gain in the mainlobe and desired NL in the nulling region.

Figure 5.  Synthesized power gain patterns for different beamwidth via linear AEP array antenna with $ {\theta _c} = {0\circ } $ and $ \gamma = - 80\;{\text{dB}} $: (a) $ {\Theta _{{\text{bw}}}} = {20\circ } $, (b) $ {\Theta _{{\text{bw}}}} = {30\circ } $, and (c) $ {\Theta _{{\text{bw}}}} = {40\circ } $.

In subsections 3.1 and 3.2, both the IEP scenarios and the AEP scenarios are simulated. Given desired NL in a nulling region, the PGPS-based algorithms always perform better than the SBPS-based algorithm. It, hence, proves that solving the PGPS optimization problem has the potential to further suppress SLL. For the PGPS-based algorithms, the proposed one can obtain lower SLL. It demonstrates that the linear approximation of the mainlobe pattern in Ref. [17] would deteriorate the performance seriously and the iterative scheme proposed in Ref. [17] can eliminate this adverse effect. In the AEP scenarios, the radiation power in the “stopband” region between the mainlobe region and the sidelobe region is relatively high compared with that in IEP scenarios. It, hence, causes a lower power gain in the mainlobe for the SBPS-based algorithm. To reduce such unnecessary power waste, one effective way is to choose some other SBPS-based solvers, however, this topic is out of the scope of this paper and would not be further discussed at the current stage.

The paper proposes a PGPS-based algorithm to further reduce SLL for the given power gain in the mainlobe and given NL in a nulling region. To deal with the non-convex PGPS problem, the proposed scheme optimizes the left-side/right-side array excitation of the power gain pattern in an iterative manner, which therefore transforms the primal problem into a convex one. Fixed the left-side excitation, the right-side excitation can be optimized with existing convex problem solvers, such as CVX and SeDumi. Then, the left-side excitation is updated accordingly with the right-side one. In such a form, the optimal solution to the primal problem is approximated. Numerical examples demonstrate that the proposed algorithm is more competitive than the existing SBPS-based and PGPS-based algorithms. Precisely, under the precondition, to obtain the same power gain $ {G_0} $ and desired NL $ \gamma $, the proposed algorithm can reduce SLL by more than 10 dB.

The authors declare no conflicts of interest.